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Solvable random-matrix ensemble with a logarithmic weakly confining potential.

Wouter Buijsman1

  • 1Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.

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Summary

This study introduces a new random matrix ensemble with a logarithmic potential, solvable using orthogonal polynomials. Its eigenvalue density follows a Lorentzian distribution, with potential applications in quantum many-body physics.

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Area of Science:

  • Mathematical Physics
  • Random Matrix Theory

Background:

  • Random matrix theory (RMT) is crucial for understanding complex quantum systems.
  • Existing RMT ensembles often lack analytical tractability for certain potentials.

Purpose of the Study:

  • To identify a novel, solvable random matrix ensemble.
  • To characterize its spectral properties and potential applications.

Main Methods:

  • The study defines a rotationally invariant random matrix ensemble with a logarithmic potential.
  • Spectral correlation functions are expressed using nonclassical Gegenbauer polynomials.
  • A sampling procedure is developed for numerical verification.

Main Results:

  • The ensemble is analytically solvable, with spectral correlation functions linked to Gegenbauer polynomials C_{n}^{(-1/2)}(x).
  • In the thermodynamic limit, the eigenvalue density is Lorentzian.
  • Numerical simulations confirm analytical findings.

Conclusions:

  • A new, analytically tractable random matrix ensemble is presented.
  • The ensemble exhibits a Lorentzian eigenvalue density and is connected to orthogonal polynomials.
  • Potential applications exist in quantum many-body physics.